Re: Kirchhoff's rules and linear dependence

[John Denker <jsd@AV8N.COM>]

"James R. Frysinger" <frysingerj@COFC.EDU> offered some
conjectures and asked for comments:

> 1. Kirchhoff's rules can uniquely solve any "flat" circuit, that is, any
> circuit that can be laid down on paper without using "crossovers" (wires
> that pass over one another without making contact).

a) I suspect that's true as stated.

b) I suspect that the same can be said for non-planar circuits.

c) I've never seen an electrical engineer approach things that
way.  Practical circuits aren't designed that way, and they
aren't analyzed that way.

> 2. Kirchhoff's rules might be able to solve non-flat circuits but there is
> no
> guarantee that they will provide enough independent equations to derive an
> exact solution.

My intuition says otherwise.  The number of loops grows faster
than the number of independent variables, so there should be
an abundance (indeed a superabundance) of equations.

> 3. For flat circuits, at least,
>         a. the number of independent node equations will equal the number of
> nodes
> minus one and
>         b. the number of independent loop equations will equal the number of
> "smallest" loops. (A "smallest" loop is one that does not contain two or
> more loops.)

That sounds about right ... in particular, for the 3x3 mesh that
M.E presented this morning, that predicts 9 loop equations.  I
think that's the right answer.  In particular, the voltage around
the "eye" in the middle is not dependent on the others, as we
can see from the following physics argument:  Using the Maxwell
equation, voltage = flux dot, imagine starting with a huge
reservoir of flux lines not threading any part of the 3x3 mesh,
and then start pushing them through one (or more) of the 8
perimeter squares into the eye, and accumulating them in the
eye.  There will be no net "flux dot" in any of the squares
except the eye.  So you can know everything there is to know
about the loop properties of the eight perimeter loops and
not know anything about the eye ... since I haven't yet told
you the rate at which flux lines are being pushed in.

If a 3x3 mesh is too much for you to visualize, the following
should suffice to illustrate the point:

       +---------+
       |\       /|
       | \     / |
       |  +---+  |
       |  |   |  |
       |  +---+  |
       | /     \ |
       |/       \|
       +---------+

... with resistors and batteries or whatever distributed on the
legs as appropriate (not shown).

Topologically it is equivalent to a cube, but it can be considered
a planar circuit, since I just drew it in the plane.  Call it a
squashed cube if you like.

I repeat that practical electronic circuits don't look like this.
Physicists are amused by such things;  most engineers are not.

Practical electronic circuits typically don't look like meshes.
Maybe from a distance they do, but if you understand them you
see they don't function that way.  Even the spectacularly
special example of an R/2R ladder with N stages can (and
should be!) analyzed without writing down N simultaneous loop
equations.

Typically engineers start with a voltage they know, and use it
to make a current they know, which in turn they use to make
another voltage they know, and so forth.

EEs are keenly aware of Kirchhoff's node law ... but they don't
call it that.  They typically don't even dignify it with a name.
They just take it for granted, or if asked about it they'll say
"the current has to go *somewhere*".

As for the loop law, it gets even less respect.  First of all,
it doesn't necessarily hold.  Ground loops and all that.  EEs
just add up enough voltage-drops to find whatever net voltage
they're interested in ... and this is very commonly less than
a whole loop's worth of voltage.  Ground wires conventionally
are not shown (even when arguably they should be) so even if
there is a loop there it doesn't look like a loop.

In more detail:  the law that EEs use says that the voltage
drop from A to B plus the voltage drop from B to C equals
the voltage drop from A to C (via B).  That is always true,
even if ABC is not a complete loop, and even under conditions
where Kirchhoff's laws do not apply (parasitic AC magnetic
fields, for instance).

===================

Bottom line:  The challenge of finding a complete set of
independent loop equations for a mesh-like circuit is
fascinating in a Martin-Gardner sort of way, but it makes
little if any contact with practical engineering.

And just to be explicitly modest:  I presently don't know how
to extend the "small loop" method to non-planar circuits.
This bugs me.  I've found some fumets, enough to make me
suspect the extension is possible, but so far it has escaped
my grasp.